Chapter VI

Valuations

Unless otherwise stated, all the rings considered in this chapter are assumed to be commutative and to possess a unit element. All the ring homomorphisms are assumed to map unit element to unit element. Every subring of a ring $A$ is assumed to contain the unit element of $A$.

If $A$ is a local ring, its maximal ideal will be denoted by $m(A)$, its residue field $A/m(A)$ will be denoted by $\kappa(A)$ and the multiplicative group of invertible elements of $A$ will be denoted by $U(A)$; then $U(A) = A - m(A)$.